Can I get a reference or some insight into the following? Suppose a particle moves by Brownian motion, starting from a point $P$ in $\mathbf{R}^{n}$. What can we say about the distribution of the time for $P$ to cross a coordinate hyperplane?
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$\begingroup$ Why do you need $\mathbb R^n$? You have to look at just one coordinate, which moves as a one-dimensional Brownian motion. And for the latter, the answer is very well-known. If I've misinterpreted the question and you are asking about the time of hitting of any coordinate hyperplane, just take a minimum of $n$ independent hitting times. $\endgroup$– zhorasterCommented Aug 29, 2015 at 14:55
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Similar to what @zhoraster points out in the comments, this can be boiled down to 1-dimensional Brownian motion because projecting the $n$-dimensional Brownian motion $X(t)$ onto the normal direction to the hyperplane gives a Brownian motion $B(t)$. So the question becomes about a the time it takes for $B(t)$ to reach a point $a\in \mathbb R$. The distribution is usually proved using the "reflection principle," which is explained in the following OCW lecture
(In particular the CDF of $T_a = \inf\{t:B(t)=a\}$ is in the 2nd line of the proof of Proposition 3)
